Nevermind. I just read somewhere that reflexive statements don't count towards symmetry. Apparently, it involves something like a diagonal class; I guess they pair this combinations in a matrix like form.

The relation R= {(1, 1), (2, 2), (3, 3), (1, 3)} is "reflexive" but not "symmetric" so reflexive does not "imply" symmetry. However, in this case there is no (x, y) in the relation without a corresponding (y, x) so this particular example is both reflexive and symmetric.